Neural networks for control: design and analysis

The universal approximation property of neural networks (NN) makes them a popular tool for approximating various functions. Therefore, it is no surprise that they also found their way in control. In control applications, NNs are commonly used to approximate complex control laws or optimal value functions (OVF) and Q-functions, which allows for simplifying optimal control problems. However, the flexibility of NNs comes with a large number of hyperparameters associated with the topology and training of the NN that need to be chosen carefully. Moreover, typical quality measures used during the training of NNs, like mean squared error, are not sufficient to ensure a safe operation of NN-based controllers. Consequently, the use of NNs in control requires the development of tailored methods that address two main challenges. The first is how to choose a suitable NN topology that allows for efficiently approximating control laws or other functions of interest in control, e.g., PWQ OVFs. The second challenge is to ensure that systems controlled by NNs can be operated safely in terms of constraint satisfaction, stability, and privacy. These requirements are unique to control applications and thus typically not considered in classical NN applications, like image or language processing. Therefore, we address these problems by developing tailored methods that enable us to derive topologies for representing continuous PWQ functions, which play a crucial role in MPC. Furthermore, by exploiting the structure of the derived topologies, we present a training method for maxout NNs that allows us to compute parameters for the NN that globally minimize the cost function considered during training. Regarding the safety of NN-based controllers, we present extensions and simplifications for analysis methods that allow us to certify stability and constraint satisfaction for linear systems based on the approximation error of the NN with respect to a stabilizing baseline controller. We also develop methods for analyzing systems controlled by NNs, based on over approximations of reachable sets, which are used to compute robustly positively invariant sets. These methods do not require a stabilizing baseline controller and can be extended for the analysis of nonlinear systems controlled by NNs. Regarding privacy, we develop a method for computing polynomial approximations that enable an efficient encrypted evaluation of NNs. In summary, the methods presented in this thesis cover a wide range of problems of NNs in control and solve or mitigate many of them. At the same time, we highlight still existing challenges, which especially include increasing the computational efficiency of the methods based on mixed-integer programming for enabling a wide use in practice.